1549380232-Automorphic_Forms_and_Applications__Sarnak_

Lecture 5. Converse theorems Let us return first to Hecke. Recall that to a modular form 00 f(T) = L ane2"inr n-l for say SL 2 ( ...

148 J.W. COGDELL, L-FUNCTIONS FOR GLn 5.1. The results Once again, let k be a global field, A its adele ring, and 'l/; a fixed n ...

LECTURE 5. CONVERSE THEOREMS 149 (2) If S '/=- 0 then II is quasi-automorphic in the sense that there is an auto- morphic repres ...

150 J.W. COGDELL, £-FUNCTIONS FOR GLn 5.2. Inverting the integral representation Let IT b e as above and let ~ E Vn be a decompo ...

LECTURE 5. CONVERSE THEOREMS 151 I(s; U€,ip') = ( W€ (~ IO ) W~,(h)I det(h)ls-(n-ml/^2 dh }Nm (A}\ GLm(A} n-m =II r w€v (hv 0 ) ...

152 J.W. COGDELL, £-FUNCTIONS FOR GLn and similarly for (JP'~"\/{, r.p')a· These are both absolutely convergent for all a and de ...

LECTURE 5. CONVERSE THEOREMS 153 Let v E S. Let Kv = GLn(ov) be the standard maximal compact subgroup of GLn(kv)· Let Pv C Ov be ...

154 J.W. COGDELL, £-FUNCTIONS FOR GLn would like to have simply that Ft.Un) = 0. Let u = (u1, ... , Un-1) E A_n-l and consider t ...

LECTURE 5. CONVERSE THEOREMS 155 The proof of this Lemma is simply an exercise in the Kirillov model of IIv and can be found in ...

156 J.W. COGDELL, £-FUNCTIONS FOR GLn 5.4.2. The case of non-empty S Let S be our non-empty set of finite places of k. Since we ...

LECTURE 5. CONVERSE THEOREMS 157 conductor of IIv. Then the ideal n = f(II) = ITv P~" Co is called the conductor of II. For each ...

158 J.W. COGDELL, £-FUNCTIONS FOR GLn Z(ks) of Gs. The requisite growth properties are satisfied and hence the map ~s t--7 <I ...

LECTURE 5. CONVERSE THEOREMS 159 it would take us too far afield to present this in this setting, Lafforgue needed this formulat ...

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Lecture 6. Converse theorems and functoriality In this section we would like to make some general remarks on how to apply these ...

162 J .W. COGDELL, £-FUNCTIONS FOR GLn lo cal Weil-Deligne group Wt or the conjectural global Langlands group L k into the Langl ...

LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 163 field k [68]. Not knowing what this should look like, one still expects to ha ...

164 J.W. COGDELL, £-FUNCTIONS FOR GLn states that to 7r and u should be associated an automorphic representation IT of GLn(A) fo ...

LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 165 an appropriate local lift 7r v f---7 IIv associating to 7r v an irreducible a ...

166 J.W. COGDELL, £-FUNCTIONS FOR GLn have been able to overcome these difficulties, as will be discussed below, and this method ...